Hyperfine structure of $S$ states in muonic deuterium

On the basis of the quasipotential method in quantum electrodynamics we calculate corrections of orders ${\alpha }^5$ and ${\alpha }^6$ to the hyperfine structure of $S\text{-wave}$ energy levels of muonic deuterium. Relativistic corrections, effects of vacuum polarization in first, second and third orders of perturbation theory, nuclear structure, and recoil corrections are taken into account. The obtained numerical values of hyperfine splitting $\Delta E^{\mathrm{HFS}}\left(1S\right)=50.2814$ meV $(1S$ state) and $\Delta E^{\mathrm{HFS}}\left(2S\right)=6.2804$ meV $(2S$ state) represent reliable estimates for a comparison with forthcoming experimental data of the CREMA Collaboration. The hyperfine structure interval ${\Delta }_{12}=8\Delta E^{\mathrm{HFS}}\left(2S\right)-\Delta E^{\mathrm{HFS}}\left(1S\right)=-0.0379$ meV can be used for a precision check of the quantum electrodynamics prediction for muonic deuterium.

Figure 1

Effects of one- and two-loop vacuum polarization in a one-photon interaction.

Figure 2

Effects of one- and two-loop vacuum polarization in second-order perturbation theory. The dashed lines denote the Coulomb photons. $\tilde{G}$ is the reduced Coulomb Green's function.

Figure 3

Effects of vacuum polarization in third-order PT. The dashed lines denote the Coulomb photons. $\tilde{G}$ is the reduced Coulomb Green's function.

Figure 4

Nuclear structure effects of order ${\alpha }^5$. The bold points denote the deuteron vertex functions.

Figure 5

Two-photon exchange amplitudes accounting for the effects of vacuum polarization and nuclear structure. The wavy lines denote the photons. The bold points denote the deuteron vertex functions.

Figure 6

Nuclear structure effects in one-photon interaction and in second-order perturbation theory. $\tilde{G}$ is the reduced Coulomb Green's function.

Figure 7

Direct two-photon exchange amplitudes with radiative corrections to the muon line giving contributions of order $\alpha {\left(Z\alpha \right)}^5$ to the HFS. The wavy lines on the diagrams denote the photons. The bold points on the diagrams denote the vertex operators of the protons or the deuterons.